diff --git a/lib/node_modules/@stdlib/math/base/special/factorialln/lib/main.js b/lib/node_modules/@stdlib/math/base/special/factorialln/lib/main.js index 192df90b8730..900805748c80 100644 --- a/lib/node_modules/@stdlib/math/base/special/factorialln/lib/main.js +++ b/lib/node_modules/@stdlib/math/base/special/factorialln/lib/main.js @@ -21,7 +21,9 @@ // MODULES // var isNegativeInteger = require( '@stdlib/math/base/assert/is-negative-integer' ); +var gamma1pm1 = require( '@stdlib/math/base/special/gamma1pm1' ); var gammaln = require( '@stdlib/math/base/special/gammaln' ); +var log1p = require( '@stdlib/math/base/special/log1p' ); // MAIN // @@ -29,6 +31,14 @@ var gammaln = require( '@stdlib/math/base/special/gammaln' ); /** * Evaluates the natural logarithm of the factorial of `x`. * +* ## Method +* +* For \\(-1 < x < 2\\), forming \\(x+1\\) can round away low-order bits of \\(x\\), which results in large relative errors near the zeros of \\(\ln \Gamma(x+1)\\) at \\(x = 0\\) and \\(x = 1\\). To avoid forming \\(x+1\\), we use the identity +* +* ```tex +* \ln \Gamma(x+1) = \operatorname{log1p}(\Gamma(x+1) - 1) = \operatorname{log1p}(\operatorname{gamma1pm1}(x)) +* ``` +* * @param {number} x - input value * @returns {number} natural logarithm of factorial of `x` * @@ -56,6 +66,9 @@ function factorialln( x ) { if ( isNegativeInteger( x ) ) { return NaN; } + if ( x > -1.0 && x < 2.0 ) { + return log1p( gamma1pm1( x ) ); + } return gammaln( x + 1.0 ); } diff --git a/lib/node_modules/@stdlib/math/base/special/factorialln/manifest.json b/lib/node_modules/@stdlib/math/base/special/factorialln/manifest.json index 48f9714d9802..a6688f495f10 100644 --- a/lib/node_modules/@stdlib/math/base/special/factorialln/manifest.json +++ b/lib/node_modules/@stdlib/math/base/special/factorialln/manifest.json @@ -38,7 +38,9 @@ "dependencies": [ "@stdlib/math/base/napi/unary", "@stdlib/math/base/assert/is-negative-integer", + "@stdlib/math/base/special/gamma1pm1", "@stdlib/math/base/special/gammaln", + "@stdlib/math/base/special/log1p", "@stdlib/constants/float64/nan" ] }, @@ -54,7 +56,9 @@ "libpath": [], "dependencies": [ "@stdlib/math/base/assert/is-negative-integer", + "@stdlib/math/base/special/gamma1pm1", "@stdlib/math/base/special/gammaln", + "@stdlib/math/base/special/log1p", "@stdlib/constants/float64/nan" ] }, @@ -70,7 +74,9 @@ "libpath": [], "dependencies": [ "@stdlib/math/base/assert/is-negative-integer", + "@stdlib/math/base/special/gamma1pm1", "@stdlib/math/base/special/gammaln", + "@stdlib/math/base/special/log1p", "@stdlib/constants/float64/nan" ] } diff --git a/lib/node_modules/@stdlib/math/base/special/factorialln/src/main.c b/lib/node_modules/@stdlib/math/base/special/factorialln/src/main.c index c8be4e611932..eb48c234675c 100644 --- a/lib/node_modules/@stdlib/math/base/special/factorialln/src/main.c +++ b/lib/node_modules/@stdlib/math/base/special/factorialln/src/main.c @@ -19,11 +19,21 @@ #include "stdlib/math/base/special/factorialln.h" #include "stdlib/constants/float64/nan.h" #include "stdlib/math/base/assert/is_negative_integer.h" +#include "stdlib/math/base/special/gamma1pm1.h" #include "stdlib/math/base/special/gammaln.h" +#include "stdlib/math/base/special/log1p.h" /** * Evaluates the natural logarithm of the factorial of `x`. * +* ## Method +* +* For \\(-1 < x < 2\\), forming \\(x+1\\) can round away low-order bits of \\(x\\), which results in large relative errors near the zeros of \\(\ln \Gamma(x+1)\\) at \\(x = 0\\) and \\(x = 1\\). To avoid forming \\(x+1\\), we use the identity +* +* ```tex +* \ln \Gamma(x+1) = \operatorname{log1p}(\Gamma(x+1) - 1) = \operatorname{log1p}(\operatorname{gamma1pm1}(x)) +* ``` +* * @param x input value * @return natural logarithm of factorial of `x` * @@ -35,5 +45,8 @@ double stdlib_base_factorialln( const double x ) { if ( stdlib_base_is_negative_integer( x ) ) { return STDLIB_CONSTANT_FLOAT64_NAN; } + if ( x > -1.0 && x < 2.0 ) { + return stdlib_base_log1p( stdlib_base_gamma1pm1( x ) ); + } return stdlib_base_gammaln( x + 1.0 ); } diff --git a/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/runner.jl b/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/runner.jl index 0d280a26bb7c..f9f9e961f39d 100644 --- a/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/runner.jl +++ b/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/runner.jl @@ -20,7 +20,7 @@ import JSON using SpecialFunctions """ - gen( x, name ) + gen( x, name ) Generate fixture data and write to file. @@ -37,13 +37,9 @@ julia> gen( x, \"data.json\" ); ``` """ function gen( x, name ) - y = Array{Float64}( undef, length( x ) ); - for i in eachindex(x) - if isinteger( x[i] ) - y[i] = lfactorial( Int64( x[i] ) ); - else - y[i] = lgamma( x[i]+1 ); - end + # Evaluate in extended precision, as forming `x+1` in double precision can round away the low-order bits of `x`: + y = setprecision( BigFloat, 2048 ) do + Float64.( loggamma.( BigFloat.( x ) .+ 1 ) ) end # Store data to be written to file as a collection: @@ -68,8 +64,20 @@ file = @__FILE__; # Extract the directory in which this file resides: dir = dirname( file ); +# Negative tiny values: +x = range( -1.0e-300, stop = -1.0e-308, length = 500 ); +gen( x, "tiny_negative.json" ); + +# Positive tiny values: +x = range( 1.0e-300, stop = 1.0e-308, length = 500 ); +gen( x, "tiny_positive.json" ); + +# Negative small values: +x = range( -0.999, stop = -0.001, length = 500 ); +gen( x, "small_negative.json" ); + # Positive small values: -x = range( 1.0, stop = 3.0, length = 1001 ); +x = range( 0.002, stop = 3.0, length = 1500 ); gen( x, "small_positive.json" ); # Positive medium values: diff --git a/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/small_negative.json b/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/small_negative.json new file mode 100644 index 000000000000..5b9a78252dc9 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/small_negative.json @@ -0,0 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/small_positive.json b/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/small_positive.json index 7cde790a1a04..9b458d336e03 100644 --- a/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/small_positive.json +++ b/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/small_positive.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/tiny_negative.json b/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/tiny_negative.json new file mode 100644 index 000000000000..45698875f505 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/tiny_negative.json @@ -0,0 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/tiny_positive.json b/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/tiny_positive.json new file mode 100644 index 000000000000..834b47900af6 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/factorialln/test/fixtures/julia/tiny_positive.json @@ -0,0 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/factorialln/test/test.js b/lib/node_modules/@stdlib/math/base/special/factorialln/test/test.js index 283fca3452bb..28e8b15facc0 100644 --- a/lib/node_modules/@stdlib/math/base/special/factorialln/test/test.js +++ b/lib/node_modules/@stdlib/math/base/special/factorialln/test/test.js @@ -36,7 +36,10 @@ var factorialln = require( './../lib' ); var veryLargePositive = require( './fixtures/julia/very_large_positive.json' ); var largePositive = require( './fixtures/julia/large_positive.json' ); var mediumPositive = require( './fixtures/julia/medium_positive.json' ); +var smallNegative = require( './fixtures/julia/small_negative.json' ); var smallPositive = require( './fixtures/julia/small_positive.json' ); +var tinyNegative = require( './fixtures/julia/tiny_negative.json' ); +var tinyPositive = require( './fixtures/julia/tiny_positive.json' ); // TESTS // @@ -55,26 +58,26 @@ tape( 'if provided a negative integer, the function returns `NaN`', function tes values = incrspace( -1.0, -500.0, -1.0 ); for ( i = 0; i < values.length; i++ ) { v = factorialln( values[ i ] ); - t.strictEqual( isnan( v ), true, 'returns NaN when provided ' + values[ i ] ); + t.strictEqual( isnan( v ), true, 'returns expected value' ); } t.end(); }); tape( 'if provided negative infinity, the function returns `NaN`', function test( t ) { var v = factorialln( NINF ); - t.strictEqual( isnan( v ), true, 'returns NaN when provided negative infinity' ); + t.strictEqual( isnan( v ), true, 'returns expected value' ); t.end(); }); tape( 'if provided positive infinity, the function returns `+infinity`', function test( t ) { var v = factorialln( PINF ); - t.ok( v, PINF, 'returns +infinity when provided +infinity' ); + t.strictEqual( v, PINF, 'returns expected value' ); t.end(); }); tape( 'if provided `NaN`, the function returns `NaN`', function test( t ) { var v = factorialln( NaN ); - t.strictEqual( isnan( v ), true, 'returns NaN when provided a NaN' ); + t.strictEqual( isnan( v ), true, 'returns expected value' ); t.end(); }); @@ -86,9 +89,8 @@ tape( 'the function evaluates the natural logarithm of the factorial of `x` (ver expected = veryLargePositive.expected; x = veryLargePositive.x; - for ( i = 0; i < x.length; i++ ) { - y = factorialln( x[i] ); + y = factorialln( x[ i ] ); t.strictEqual( isAlmostSameValue( y, expected[ i ], 1 ), true, 'returns expected value' ); } t.end(); @@ -102,9 +104,8 @@ tape( 'the function evaluates the natural logarithm of the factorial of `x` (lar expected = largePositive.expected; x = largePositive.x; - for ( i = 0; i < x.length; i++ ) { - y = factorialln( x[i] ); + y = factorialln( x[ i ] ); t.strictEqual( isAlmostSameValue( y, expected[ i ], 1 ), true, 'returns expected value' ); } t.end(); @@ -119,12 +120,27 @@ tape( 'the function evaluates the natural logarithm of the factorial of `x` (med expected = mediumPositive.expected; x = mediumPositive.x; for ( i = 0; i < x.length; i++ ) { - y = factorialln( x[i] ); + y = factorialln( x[ i ] ); t.strictEqual( isAlmostSameValue( y, expected[ i ], 1 ), true, 'returns expected value' ); } t.end(); }); +tape( 'the function evaluates the natural logarithm of the factorial of `x` (small negative values)', function test( t ) { + var expected; + var x; + var y; + var i; + + expected = smallNegative.expected; + x = smallNegative.x; + for ( i = 0; i < x.length; i++ ) { + y = factorialln( x[ i ] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 4 ), true, 'returns expected value' ); + } + t.end(); +}); + tape( 'the function evaluates the natural logarithm of the factorial of `x` (small positive values)', function test( t ) { var expected; var x; @@ -134,13 +150,43 @@ tape( 'the function evaluates the natural logarithm of the factorial of `x` (sma expected = smallPositive.expected; x = smallPositive.x; for ( i = 0; i < x.length; i++ ) { - y = factorialln( x[i] ); + y = factorialln( x[ i ] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 3 ), true, 'returns expected value' ); + } + t.end(); +}); + +tape( 'the function evaluates the natural logarithm of the factorial of `x` (tiny negative values)', function test( t ) { + var expected; + var x; + var y; + var i; + + expected = tinyNegative.expected; + x = tinyNegative.x; + for ( i = 0; i < x.length; i++ ) { + y = factorialln( x[ i ] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 1 ), true, 'returns expected value' ); + } + t.end(); +}); + +tape( 'the function evaluates the natural logarithm of the factorial of `x` (tiny positive values)', function test( t ) { + var expected; + var x; + var y; + var i; + + expected = tinyPositive.expected; + x = tinyPositive.x; + for ( i = 0; i < x.length; i++ ) { + y = factorialln( x[ i ] ); t.strictEqual( isAlmostSameValue( y, expected[ i ], 1 ), true, 'returns expected value' ); } t.end(); }); -tape( 'the function returns a value almost equal to `ln( factorial( x )` for `x` in `[0,171]', function test( t ) { +tape( 'the function returns a value almost equal to `ln( factorial( x ) )` for `x` in `[0,171]', function test( t ) { var expected; var values; var v; @@ -150,7 +196,7 @@ tape( 'the function returns a value almost equal to `ln( factorial( x )` for `x` for ( i = 0; i < values.length; i++ ) { v = factorialln( values[ i ] ); expected = ln( factorial( values[ i ] ) ); - t.strictEqual( isAlmostSameValue( v, expected, 6 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( v, expected, 7 ), true, 'returns expected value' ); } t.end(); }); diff --git a/lib/node_modules/@stdlib/math/base/special/factorialln/test/test.native.js b/lib/node_modules/@stdlib/math/base/special/factorialln/test/test.native.js index 6a82c22b0a3f..ca86d2c13ede 100644 --- a/lib/node_modules/@stdlib/math/base/special/factorialln/test/test.native.js +++ b/lib/node_modules/@stdlib/math/base/special/factorialln/test/test.native.js @@ -37,7 +37,10 @@ var tryRequire = require( '@stdlib/utils/try-require' ); var veryLargePositive = require( './fixtures/julia/very_large_positive.json' ); var largePositive = require( './fixtures/julia/large_positive.json' ); var mediumPositive = require( './fixtures/julia/medium_positive.json' ); +var smallNegative = require( './fixtures/julia/small_negative.json' ); var smallPositive = require( './fixtures/julia/small_positive.json' ); +var tinyNegative = require( './fixtures/julia/tiny_negative.json' ); +var tinyPositive = require( './fixtures/julia/tiny_positive.json' ); // VARIABLES // @@ -64,26 +67,26 @@ tape( 'if provided a negative integer, the function returns `NaN`', opts, functi values = incrspace( -1.0, -500.0, -1.0 ); for ( i = 0; i < values.length; i++ ) { v = factorialln( values[ i ] ); - t.strictEqual( isnan( v ), true, 'returns expected value when provided ' + values[ i ] ); + t.strictEqual( isnan( v ), true, 'returns expected value' ); } t.end(); }); tape( 'if provided negative infinity, the function returns `NaN`', opts, function test( t ) { var v = factorialln( NINF ); - t.strictEqual( isnan( v ), true, 'returns expected value when provided negative infinity' ); + t.strictEqual( isnan( v ), true, 'returns expected value' ); t.end(); }); tape( 'if provided positive infinity, the function returns `+infinity`', opts, function test( t ) { var v = factorialln( PINF ); - t.ok( v, PINF, 'returns +infinity when provided +infinity' ); + t.strictEqual( v, PINF, 'returns expected value' ); t.end(); }); tape( 'if provided `NaN`, the function returns `NaN`', opts, function test( t ) { var v = factorialln( NaN ); - t.strictEqual( isnan( v ), true, 'returns expected value when provided a NaN' ); + t.strictEqual( isnan( v ), true, 'returns expected value' ); t.end(); }); @@ -95,9 +98,8 @@ tape( 'the function evaluates the natural logarithm of the factorial of `x` (ver expected = veryLargePositive.expected; x = veryLargePositive.x; - for ( i = 0; i < x.length; i++ ) { - y = factorialln( x[i] ); + y = factorialln( x[ i ] ); t.strictEqual( isAlmostSameValue( y, expected[ i ], 1 ), true, 'returns expected value' ); } t.end(); @@ -111,9 +113,8 @@ tape( 'the function evaluates the natural logarithm of the factorial of `x` (lar expected = largePositive.expected; x = largePositive.x; - for ( i = 0; i < x.length; i++ ) { - y = factorialln( x[i] ); + y = factorialln( x[ i ] ); t.strictEqual( isAlmostSameValue( y, expected[ i ], 1 ), true, 'returns expected value' ); } t.end(); @@ -128,7 +129,7 @@ tape( 'the function evaluates the natural logarithm of the factorial of `x` (med expected = mediumPositive.expected; x = mediumPositive.x; for ( i = 0; i < x.length; i++ ) { - y = factorialln( x[i] ); + y = factorialln( x[ i ] ); // NOTE: the tolerance here is larger than for the JavaScript implementation due to compiler optimizations which may be performed resulting in result divergence. For discussion, see https://github.com/stdlib-js/stdlib/pull/2774#discussion_r1712904184 t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' ); @@ -136,6 +137,21 @@ tape( 'the function evaluates the natural logarithm of the factorial of `x` (med t.end(); }); +tape( 'the function evaluates the natural logarithm of the factorial of `x` (small negative values)', opts, function test( t ) { + var expected; + var x; + var y; + var i; + + expected = smallNegative.expected; + x = smallNegative.x; + for ( i = 0; i < x.length; i++ ) { + y = factorialln( x[ i ] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 4 ), true, 'returns expected value' ); + } + t.end(); +}); + tape( 'the function evaluates the natural logarithm of the factorial of `x` (small positive values)', opts, function test( t ) { var expected; var x; @@ -145,13 +161,43 @@ tape( 'the function evaluates the natural logarithm of the factorial of `x` (sma expected = smallPositive.expected; x = smallPositive.x; for ( i = 0; i < x.length; i++ ) { - y = factorialln( x[i] ); + y = factorialln( x[ i ] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 3 ), true, 'returns expected value' ); + } + t.end(); +}); + +tape( 'the function evaluates the natural logarithm of the factorial of `x` (tiny negative values)', opts, function test( t ) { + var expected; + var x; + var y; + var i; + + expected = tinyNegative.expected; + x = tinyNegative.x; + for ( i = 0; i < x.length; i++ ) { + y = factorialln( x[ i ] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 1 ), true, 'returns expected value' ); + } + t.end(); +}); + +tape( 'the function evaluates the natural logarithm of the factorial of `x` (tiny positive values)', opts, function test( t ) { + var expected; + var x; + var y; + var i; + + expected = tinyPositive.expected; + x = tinyPositive.x; + for ( i = 0; i < x.length; i++ ) { + y = factorialln( x[ i ] ); t.strictEqual( isAlmostSameValue( y, expected[ i ], 1 ), true, 'returns expected value' ); } t.end(); }); -tape( 'the function returns a value almost equal to `ln( factorial( x )` for `x` in `[0,171]', opts, function test( t ) { +tape( 'the function returns a value almost equal to `ln( factorial( x ) )` for `x` in `[0,171]', opts, function test( t ) { var expected; var values; var v; @@ -161,7 +207,7 @@ tape( 'the function returns a value almost equal to `ln( factorial( x )` for `x` for ( i = 0; i < values.length; i++ ) { v = factorialln( values[ i ] ); expected = ln( factorial( values[ i ] ) ); - t.strictEqual( isAlmostSameValue( v, expected, 6 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( v, expected, 7 ), true, 'returns expected value' ); } t.end(); });